LEARNING DISCOURSE pptx

LEARNING DISCOURSE
Discursive approaches to research
in mathematics education
Edited by
CAROLYN KIERAN
ELLICE FORMAN
and
ANNA SFARD
This book was previously published as a PME Special Issue in Educational Stud-
ies in Mathematics, Vol. 46 (1–3), 2001, under the title : BRIDGING THE INDI-
VIDUAL AND THE SOCIAL: DISCURSIVE APPROACHES TO RESEARCH
IN MATHEMATICS EDUCATION
KLUWER ACADEMIC PUBLISHERS
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Print ISBN: 1-4020-1024-9
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TABLE OF CONTENTS
Guest Editorial

1-11
Acknowledgements
ANNA SFARD / There is more to discourse than meets the ears:
Looking at thinking as communicating to learn more about
mathematical learning
12
13–57
BERT VAN OERS / Educational forms of initiation in mathematical
culture
59–85
STEPHEN LERMAN / Cultural, discursive psychology: A sociocul-
tural approach to studying the teaching and learning of
mathematics
87–113
ELLICE FORMAN and ELLEN ANSELL /The multiple voices of a
mathematics classroom community
115–142
MARY CATHERINE O’CONNOR / “Can any fraction be turned
into a decimal?” A case study of a mathematical group
discussion
143–185
CAROLYN KIERAN / The mathematical discourse of 13-year-old
partnered problem solving and its relation to the mathem-
atics that emerges
187–228
VICKI ZACK and BARBARA GRAVES / Making mathematical
meaning through dialogue: “Once you think of it, the Z
minus three seems pretty weird”
229–271
Commentary papers

CELIA HOYLES / From describing to designing mathematical activ-
ity: The next step in developing a social approach to
research in mathematics education?
273–286
FALK SEEGER / Research on discourse in the mathematics
classroom: A commentary
287–297
This page intentionally left blank
GUEST EDITORIAL
LEARNING DISCOURSE: SOCIOCULTURAL APPROACHES TO
RESEARCH IN MATHEMATICS EDUCATION
While looking at the papers collected in this volume one feels that, in spite
of their diverse themes, these seven studies have quite a lot in common
and, as a collection, seem to be signaling the existence of a distinct, re-
latively new type of research in mathematics education. A comparison
with, say, a fifteen-year-old issue of Educational Studies in Mathema-
tics or of Journal for Research in Mathematics Education would reveal
a long series of differences. To begin with, the present articles simply
look different from their older counterparts: They are longer and have a
highly variable format, often not even remotely reminiscent of the clas-
sical background-method-sample-findings-discussion structure that reigns
in the former research reports. Long segments of conversation transcripts
take the place of the once ubiquitous graphs and tables. As we start read-
ing, we discover substantial differences in vocabulary. The language of
mental schemes, misconceptions, and cognitive conflicts seems to be giving
way to a discourse on activities, patterns of interaction, and communica-
tion failures. While the older texts speak of learning in terms of personal
acquisition, the newer ones portray it as the process of becoming a parti-
cipant in a collective doing. And last but not least, the classroom scenes
that we see as we go on reading have very little in common with what

we got used to in the older papers. To be sure, finding a detailed descrip-
tion of a learning activity in a research paper was a rare occurrence until
recently, and in the majority of cases we had to rely on our own exper-
ience while trying to imagine the life of the class in which the authors
conducted their study. In spite of this limitation, much can be said also
about the differences in the ways of learning investigated in the two types
of research: The traditional mathematical classroom featuring one black-
board, one outspoken teacher and twenty to forty silent students seems to
belong to history.
1
It has been replaced by small teams of learners talking
to each other, by groups of students voicing their opinions in whole class
discussions, and by children and grownups grappling with mathematical
problems in real-life situations.
Educational Studies in Mathematics 46: 1–12, 2001.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
2
CAROLYN KIERAN, ELLICE FORMAN, AND ANNA SFARD
All these innovative features, when taken together, seem to make a real
difference and to define a distinctive research framework that, because of
its obvious emphasis on the issues of language and communication, can be
called discursive or communicational. To be sure, this special framework,
although quite widespread and increasingly popular these days, is still un-
der construction. And yet, considering the progress that has already been
made, the time seems ripe for an intermediary summary and reflection.
The aim of this special issue is to put discursive research in the limelight
and to spur some thinking about the reasons for its appearance, about its
nature, and about its possible advantages and pitfalls. Let us now address
these issues briefly, one by one.
The first question to ask concerns the reasons for the advent of the dis-

cursive approach. To give a proper answer, one has to take a broader look
at the history, and not just of mathematics education, but of research on hu-
man thinking in general. This latter research is, and always has been, torn
between two complementary, but not necessarily compatible, goals. On
the one hand, the intention of the researcher is to fathom the phenomenon
of human thinking in all its uniqueness and with all its ramifications. On
the other hand, the method employed must be rigorous enough to put this
research on a par with any other scientific endeavor with respect to co-
gency, trustworthiness, and, above all, usefulness. These two goals create
an essential tension that fuels the incessant change. While the request of
scientificity (whatever this term means at a given moment) pushes to-
ward simplicity and feeds the belief in cross-contextual invariants, the
wish for an all-encompassing, true-to-life picture of human cognitive activ-
ities implies that the formidable complexity of the phenomenon should
never disappear from the researcher’s sight. No wonder then, that the relat-
ively brief history of cognitive studies is stormy and replete with dramatic
turnabouts.
On the face of it, the main question that needs to be answered before
the dilemma of the conflicting goals can be solved is that of the proper
method of inquiry. And yet, not in many fields of research is the way of
conceptualizing the object of investigation more sensitive to methodolo-
gical issues than in the study of the human mind. Judging from history, the
uncompromising insistence on methodological rigor, especially if gauged
according to criteria borrowed from the ‘exact’ sciences, forces research-
ers to bend, and eventually forget, the original focus of their endeavor.
This is what happened when behaviorists decided to purge psychological
discourse of any reference to mental non-observables, and this is what
happened again not long after the advent of computer science, when tech-
nology brought back the hope of a truly scientific insight into the workings
GUEST EDITORIAL

3
of the human mind. Jerome Bruner, one of the founding fathers of the
‘cognitive revolution’ of the late 1950s admits that his and his colleagues’
‘all-out effort to establish meaning as the central concept of psychology…’
(Bruner, 1990, p. 2), grounded in the computer metaphor of mind, did
not achieve its goal and, in consequence, failed to deliver on its prom-
ise of groundbreaking insights into the specificity of the human intellect.
As the recent proliferation of critical publications makes clear, also Pia-
get’s impressive attempt to meet the challenge of the conflicting goals
by modeling the development of human thinking on Darwinian theory of
evolution proved unsatisfactory in many respects (see e.g. Bruner, 1985).
The insufficiency of all these approaches expressed itself, among others, in
their inability to bring about a lasting betterment of the human condition,
which is the ultimate goal of any scientific endeavor. Thus, for example,
none of the theories produced by the different frameworks could account
in a satisfactory way for such phenomena as the persistent failure of many
students in mathematics or the stubborn irreproducibility of educational
success.
The first notions about possible reasons for this pervasive difficulty
came following a wave of cognition-oriented cross-cultural studies that
began in the early 1920s. At that time, psychologists and educators from
diverse scientific traditions began arriving in cultures far removed from
their own, convinced that “[i]n the realm of culture, outsideness is a most
powerful factor in understanding” (Bakhtin, 1986, p.7) and keen to observe
what came to be known as higher psychological processes cast against the
background of foreign traditions (for an historical survey see Cole, 1996).
Mathematical thinking, considered as a paradigmatic example of such a
process, and as one that is particularly liable to rigorous investigation, be-
came the preferred object of study.
2

The guiding assumption of the early
studies was that this uniquely human form of cognitive activity may be
found in pre-industrial cultures in their nascent, underdeveloped form. By
watching the incipient editions of these processes, psychologists hoped to
learn about the cultural invariants of human cognition. And yet, as it soon
became clear, venturing into unfamiliar cultural settings to look for phe-
nomena defined according to one’s own cultural heritage is an inherently
problematic, ultimately misguided, endeavor. Initially, doubts were raised
about the methods of study. The traditional forms of experimental design
became questioned when the experimenters realized that school mathemat-
ical problems, imported directly from the researchers’ own culture, would
only too often turn out to be completely foreign to the respondent. This
clearly created the possibility of major misinterpretations, with the invest-
4
CAROLYN KIERAN, ELLICE FORMAN, AND ANNA SFARD
igators conferring on their findings meanings dictated by their own cultural
background (Cole, 1996).
Increasingly suspicious about the experimental method, some of the re-
searchers began supplementing their investigations with descriptive studies
in which the focus shifted from laboratory problem solving to spontan-
eous everyday activities. A long series of research projects devoted to
what came to be known as everyday, street, workplace or supermarket
mathematics followed. The main merit of all these studies was that they
obviated the need for the researcher’s regulatory intervention, at least in
the initial phase of the investigation that was usually carried out as an
ethnographical observation. An experimental study would then often be
devised so as to make it possible for the subjects to communicate with the
researcher on their own terms. The change of approach proved itself when
the non-interventional studies began producing results dramatically differ-
ent from those one would expect on the grounds of the subjects’ former

performance on school tasks purported to involve ‘the same’ cognitive
functions. It soon became clear that the superior everyday mathematical
performance of people who tended to fail on school tasks is not an acci-
dental, isolated phenomenon. What was found among Kpelle rice sellers
in the mid-1960s (Cole, 1996) was observed over and over again among
Vai tailors (Reed and Lave, 1979), Brazilian street vendors (Saxe, 1991;
Nunes et al., 1993), dairy warehouse workers (Scribner, 1983/1997), and
American weight-watchers and shoppers (Lave, 1988).
At this point the methodological doubt turned epistemological. Psy-
chologists started questioning what until now had been taken for granted
even without being explicitly spelled out. A common denominator of all
the traditional approaches to thinking was the vision of mind as a ‘mirror
of nature’ (Rorty, 1979) – a container to be filled with reflections of, or
structures residing in, the external world. Whether simply received or in-
dividually constructed, it was believed that these structures – known as
knowledge, concepts, or mental schemes – were regulated by universal
external factors, and should thus be more or less the same for all human
beings. Once acquired, each such structure should lead to similar behaviors
in all the situations in which this structure could be identified. Similarly,
the cognitive processes that produced and used these entities were expec-
ted to be cross-contextually invariant, that is, governed by universal rules
that remain basically the same across different social, cultural, historical
and situational settings. Those who were taking a closer look at cognition
across cultural and situational boundaries could not help wondering about
the soundness of this assumption, or at least about its testability. Sooner
or later, this essential doubt would force them to question the concep-
GUEST EDITORIAL
5
tual foundations of the traditional framework. This is how the acquisition
metaphor, upon which the time-honored cognitivist approach was resting,

became the primary suspect.
To this very day, the acquisitionist framework, its impressive history
notwithstanding, is a target of criticism coming from the somewhat eclectic
group of thinkers who are often called sociocultural. In fact, many different
names have been given to this rich and diverse cluster of approaches.
3
What sets these approaches apart as a distinct group is the fact that most
of them are associated with the Vygotskian school of thought, and that
they all promote the vision of human thinking as essentially social in its
origins and as inextricably dependent on historical, cultural, and situational
factors. It is important to stress that our historical account by no means
exhausts the list of approaches that can be called sociocultural, nor does
it cover all the events that led to the advent of this variegated trend.
4
In
our selective and, of necessity, very brief survey we have focused on those
developments that had a direct bearing on cognitive research in general,
and on research in mathematical education, in particular.
The discursive approach announced in the title of this special issue can
be viewed as one of many possible implementations of the sociocultural
call for research that acknowledges the inherently social nature of human
thought. Not all the contributors to this volume are using the name ‘dis-
cursive’ and some of them may eschew any explicit descriptions of the
epistemological and ontological underpinnings of their research. Never-
theless, a number of theoretical assumptions can be identified that seem
to be guiding all the authors. These overarching foundational motifs are
what defines the discursive framework. The reader will come across the
common theoretical threads while reading the papers. The articles by van
Oers, by Lerman, and by Sfard, which all deal with the conceptual in-
frastructure of the discursive research explicitly, will help in revealing

these common threads. At this point, suffice it to say that within the dis-
cursive framework, thinking is conceptualized as a special case of the
activity of communication and learning mathematics means becoming flu-
ent in a discourse that would be recognized as mathematical by expert
interlocutors. As will be explained by the contributors themselves, these
deceptively simple definitions turn out to have quite far reaching theor-
etical and practical entailments. In the remainder of this introduction, let
us limit ourselves to the question of how the discursive approach helps in
resolving the dilemmas that have been challenging our research and fueling
its incessant change ever since its earliest beginnings.
Let us start with the question of whether the discursive approach stands
a good chance of capturing what is unique in human thinking. The first
6
CAROLYN KIERAN, ELLICE FORMAN, AND ANNA SFARD
thing to note is that while the more traditional frameworks conceptualize
learning as intellectual acquisition, and thus as a change in the individual
learner, the discursive approach focuses on the change in one’s ways of
communicating with others. This complicates the picture and makes it
much richer. While the place of the individual is not denied, it is concep-
tualized in a whole new way. No longer is the individual learner viewed
as the only object of change; furthermore, the change itself is no longer
regarded as stand-alone and independent of that which affects the com-
munity of learners as a whole. Indeed, the vision of learning as becoming
a participant in a practice must lead to the conclusion that in this process,
the practice itself is bound to undergo modifications. Thus, the inclusion
of the community in the picture of learning affects the scope of things
that must be considered when the change in the activities of an individual
learner is studied.
When regarded not as an isolated entity but as a part of a larger whole,
the learner becomes but an inextricable element of a new, much broader

unit of analysis, many ingredients of which must be brought into the ac-
count even if the ultimate focus of study is change in the individual learner’s
activities. More specifically, when learning mathematics is conceptualized
as developing a discourse, probably the most natural units of analysis can
be found in the discourse itself (as opposed to such formerly favored units
as concepts, mental schemes, or student’s knowledge).
Indeed, the focus of the studies reported in this volume is on the dis-
course generated by students grappling with mathematical problems. Thus,
it is interesting to see how the classroom conversation develops on both the
collective and individual level when the group of children in O’Connor’s
study responds to the teacher’s challenging question “Can any fraction be
turned into a decimal?” O’Connor examines the fit between the math-
ematical content (rational numbers and their representational forms) and
a whole class position-driven discussion in an upper elementary school
classroom. Position-driven discussions occur when a teacher orchestrates
an argument among a group of students of one conceptually challenging
central question with a limited number of options. Like O’Connor, Forman
and Ansell investigate how a teacher orchestrates the discourse in her ele-
mentary school classroom. Unlike O’Connor, however, they argue that
voices from the past, present, and future, and from outside as well as
inside the classroom walls, animate the discussion of students’ strategies
for solving multi-digit word problems. These voices come from the stu-
dents’ families and the teacher’s educational experiences; they represent
the memories, attitudes, emotions, and expectations about traditional and
reform educational practices in mathematics.
GUEST EDITORIAL
7
While the conversations in both of these articles involve the whole class
and are orchestrated by the teacher, the study in the Zack and Graves article
looks at the discourse of groups of students in problem-solving situations.

The particular focus is the way in which the differences among the posi-
tions of the participants function and how they enable the learners to jointly
construct new knowledge. Similarly, the studies in Kieran’s and Sfard’s
articles offer a glimpse into dyadic peer interactions. Kieran explores the
emergence of collective mathematical thinking and the ways in which the
mathematical discourse of some individuals changes as a result of the
group experience. Sfard, on the other hand, tries to fathom the nature and
the reasons for the evident ineffectiveness of an interaction between two
students who try to solve a mathematical problem.
It must be immediately stressed that discourse is not the only pos-
sible source of units of analysis for sociocultural research, nor even the
only one considered by the contributors to this special issue. Among the
most widely known alternatives are activity, the unit proposed by Vygot-
skian scholars who call themselves activity theorists (Leont’ev, 1978; En-
gestrom, 1987),
5
culture, as preferred by at least some of cultural psycholo-
gists who view learning as enculturation (Tomasello, 1999); and practice,
introduced by those among sociocultural thinkers who are most strongly
oriented toward sociological issues (Lave, 1988; Lave and Wenger, 1991;
Wenger, 1998).
A review of these other possibilities and the explanation of their rel-
ative advantages can be found in the article by van Oers, who organizes
his exposition around the fundamental question “What is really mathem-
atical?”. He provides an historical overview of research on mathematics
learning in classroom settings before articulating the discursive approach.
Building upon the theories of Vygotsky and Bakhtin, van Oers outlines
an emerging framework for future research in which notions of activity,
practice, and discourse play prominent roles. Lerman also surveys a variety
of theories that have influenced mathematics education and provides his

own version of cultural, discursive psychology. In this survey, he discusses
discursive psychology, cultural psychology, and sociocultural research, in
order to work towards a synthesis. In contrast, Sfard makes a clear choice
and argues for the advantages of the framework that takes discourse and
communication as its pivotal concepts.
Whether one speaks about learning in terms of discourse, activity, cul-
ture, or practice, the focus is on the change generated by interpersonal
interactions, and this, as has already been mentioned, results in a picture
which is more complex and closer to life than in the traditional cognitivist
studies. The question that now begs to be asked is whether all this rich-
8
CAROLYN KIERAN, ELLICE FORMAN, AND ANNA SFARD
ness does not come at the expense of the scientific elegance, cogency, and
trustworthiness of the research. While looking at recent publications that
can count as sociocultural, one may say that, indeed, we are still paying
the methodological cost of the decision to put a premium on the goal of
capturing the intricacies of learning in all their specificity and uniqueness.
We must realize that when it comes to tools and techniques that would
match this endeavor, we have yet a long way to go. Unlike in the former,
positivist era, we now have to craft our ways of analyzing data each time
anew, appropriately to the questions we are asking, and in accord with the
data we were able to collect.
This does not mean, however, that it is not possible to build a basic
reservoir of sound methodological tools. With its well-defined, directly
accessible object of study, the discursive approach seems to be on its way
to becoming a fully-fledged research framework, complete with a set of
reliable methods of data analysis. In the last years, many impressive meth-
odological advances have been made within this area. In addition to the
general-purpose techniques, such as those gathered under the names con-
versation analysis and discourse analysis, numerous new tools specially

crafted to fit the particular needs of the research in mathematics education
are appearing these days with an increasing frequency.
An assortment of such methods may be found in this special issue.
O’Connor, who comes to mathematics education from the field of applied
linguistics, has always made extensive use of the methods of discourse ana-
lysis in her studies on interaction patterns in mathematical classrooms. In
the paper included in this volume, she builds her own techniques of looking
while trying to capture the ways in which mathematical content evolves as
a result of interaction. In her detailed account, O’Connor shows us how
the whole-class discussion unfolds, helping us understand the conceptual,
pedagogical, and interpersonal dilemmas that emerge during discussions of
challenging mathematical content. She uses units that parse the argument
into claims and counterclaims with supporting evidence. She also identifies
units that illustrate the teacher’s skill at managing conceptual and commu-
nicative confusion. Forman and Ansell employ a hierarchy of units in their
analysis from the molar (such as a lesson) to the molecular (sequences of
talk about a particular topic). Furthermore, they examine critical junctures
or changes in the structure of events, which may allow one to make infer-
ences about participants’ interpretations of those events. Zack and Graves
structure the extract that they present into four parts, which emerge as a
function of participants’ differing roles and stances during the mathemat-
ical interaction. This structuring device affords an analysis of the process
whereby individual and group developmental trajectories are constructed,
GUEST EDITORIAL
9
as well as an exploration of the relationship between discourse and know-
ing. Kieran uses an interactivity flowchart, adapted from an earlier study
for which it was created (cf., Sfard and Kieran, 2001), to segment the
discourse of participants into personal and interpersonal channels of talk.
With a focus directed toward those object-level utterances that move the

mathematical dimensions of the discourse forward, Kieran hypothesizes
why there might be discrepancies between partners in their subsequent
individual work. In her attempt to understand the nature of and reasons for
the observed communication failure, Sfard applies the interactivity flow-
chart along with another type of analysis developed in her former study
with Kieran: She follows the course of the mathematical conversation with
the help

of
focal analysis – a method that aims at ‘mapping the trajectory’
of the object of conversation.
In this special issue, the complexity of the phenomena under study is
reflected in the multi-level analyses of the discourse. All the authors are
discussing the development of mathematical communication, and while
doing so, they are alternating between the analysis of students’ single turns
and the examination of patterns to be found in sequences of thematically
connected utterances. This may be compared to the study of the mech-
anics of water where, at some points, the researchers may be watching
regularities in the movement of individual particles, and at other times
may choose to investigate the geometry and periodic recurrence of waves
and whirls. The macro- and micro-level pictures obtained in these ways
do not resemble each other, and yet, both are needed by those who try
to understand the complex phenomenon under study. In the same vein,
whatever the particular focus or level of analysis in the studies presented
in this volume, the phenomenon under study remains the same: All the
authors are looking at classroom communication that evolves so as to be-
come genuinely mathematical and to allow for solving problems that were
intractable within other discourses.
A message similar to the one conveyed by the above comparison can be
found in the ‘zoom of lens’ metaphor invoked in this volume by Lerman

to explain the relation between the individual and social research perspect-
ives. The much debated split between these two perspectives is referred to
in the title of this special issue, ‘Bridging the Individual and the Social’.
This split has been worrying researchers for some time now. The seemingly
incompatible perspectives are producing two incomplete types of studies,
each of which is ‘telling only half of the good story’ (Cobb, 1996). The
call for bridging the two halves follows. We turned this call into the title
for the special issue, but not necessarily because we believe that bridging
is what needs to be done. Rather, we used the slogan because it points to
10
CAROLYN KIERAN, ELLICE FORMAN, AND ANNA SFARD
the dilemma that seems to be still at the center of researchers’ attention.
Our solution to this dilemma is to deconstruct the dichotomy, and not to
unify the halves. Indeed, as the water-study metaphor makes clear, by de-
fining thinking as communicating we are sidestepping the split rather then
bridging a gap. The problematic dichotomy between the individual and
social research perspectives is no longer an issue when one realizes that
the cognitivist (‘individualistic’) and interactionist (‘social’) approaches
are but two ways of looking at what is basically one and the same phe-
nomenon: the phenomenon of communication, one that originates between
people and does not exist without the collective even if it may temporarily
involve only one interlocutor. The social nature of the individual is the
principal message of this special issue.
N
OTES
1.
2.
3.
4.
5.

Even if this is not true for many mathematics classrooms around us, it certainly is true
for those in which researchers nowadays choose to conduct their studies.
The reasons for the particular appropriateness of mathematics for uncovering factors
contributing to human cognitive development are eloquently spelled out by Reed and
Lave (1979) in the paper with the telling title Arithmetic as a tool for investigating
relations between culture and cognition.
Several different terms have been used to characterize the school of thought that began
with Vygotsky: sociocultural, cultural-historical activity theory, cultural psychology,
neo-Vygotskian. Vygotsky himself used the term cultural-historical (Cole, 1995; v an
Oers, 1998).
Among the thinkers whose work had a decisive influence on this development one
should mention, above all, the Austrian-British philosopher Ludwig Wittgenstein whose
seminal work on language brought the issue of communication to the center of psycho-
logical research; and of the American and German social philosophers George Herbert
Mead and Alfred Schutz, who stressed, each one of them in his own way, the tight
relations between human thought and social interactions. (See Valsiner and van de
Veer, 2000, and Cole, 1996, for historical overviews of the relevant theories.)
See
also
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GUEST EDITORIAL 11
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ANNA SFARD, ELLICE FORMAN AND CAROLYN KIERAN
12
CAROLYN KIERAN, ELLICE FORMAN, AND ANNA SFARD
ACKNOWLEDGEMENTS
We, the guest editors, are pleased to have had the opportunity to produce
this Special Issue, one of the series emanating from the International Group
for the Psychology of Mathematics Education (PME). However, we hope
that this Special Issue will be of interest not only to PME researchers, but
also to the broader scientific community concerned with issues related to
mathematical discourse and communication.
The Special Issue would not have been possible without the collabora-
tion and cooperation of several individuals. We especially wish to thank the
authors of the seven main papers who share their research with readers of
this volume. Their theoretical discussions and analyses touch upon crucial
aspects of discursive interactions in the mathematics classroom. We are
also grateful to Celia Hoyles and Falk Seeger for their contributions in the
form of commentary papers. Among the issues raised for consideration,
Celia emphasizes in particular the importance of tool mediation and the
design of mathematical activities, while Falk argues for more long-term
studies on the formation of proficient discursive classrooms.
We want to acknowledge, as well, the work done by the reviewers of
the research papers. Their timely, thorough, and insightful comments were
greatly appreciated by the authors. We express our gratitude to the Kluwer

editorial staff for their patient and expert handling of the various stages of
the development of this Special Issue. Last, but not least, we owe special
thanks to Heinz Steinbring who, as shadow editor, shepherded this volume
to its completion and helped us to deal with several matters along the way.
A warm thank-you to all!
ANNA SFARD
THERE IS MORE TO DISCOURSE THAN MEETS THE EARS:
LOOKING AT THINKING AS COMMUNICATING TO LEARN
MORE ABOUT MATHEMATICAL LEARNING
ABSTRACT. Traditional approaches to research into mathematical thinking, such as the
study of misconceptions and tacit models, have brought significant insight into the teaching
and learning of mathematics, but have also left many important problems unresolved. In
this paper, after taking a close look at two episodes that give rise to a number of difficult
questions, I propose to base research on a metaphor of thinking-as-communicating. This
conceptualization entails viewing learning mathematics as an initiation to a certain well
defined discourse. Mathematical discourse is made special by two main factors: first,
by its exceptional reliance on symbolic artifacts as its communication-mediating tools,
and second, by the particular meta-rules that regulate this type of communication. The
meta-rules are the observer’s construct and they usually remain tacit for the participants
of the discourse. In this paper I argue that by eliciting these special elements of math-
ematical communication, one has a better chance of accounting for at least some of the
still puzzling phenomena. To show how it works, I revisit the episodes presented at the
beginning of the paper, reformulate the ensuing questions in the language of thinking-as-
communication, and re-address the old quandaries with the help of special analytic tools
that help in combining analysis of mathematical content of classroom interaction with
attention to meta-level concerns of the participants.
In the domain of mathematics education, the term discourse seems these
days to be on everyone’s lips. It features prominently in research papers,
it can be heard in teacher preparation courses, and it appears time and
again in a variety of programmatic documents that purport to establish

instructional policies (see e.g. NCTM, 2000). All this could be interpreted
as showing merely that we became as aware as ever of the importance
of mathematical conversation for the success of mathematical learning. In
this paper, I will try to show that there is more to discourse than meets the
ears, and that putting communication in the heart of mathematics education
is likely to change not only the way we teach but also the way we think
about learning and about what is being learned. Above all, I will be arguing
that communication should be viewed not as a mere aid to thinking, but as
almost tantamount to the thinking itself. The communicational approach
to cognition, which is under scrutiny in this paper, is built around this basic
theoretical principle.
In what follows, I present the resulting vision of learning and explain
why this conceptualization can be expected to make a significant con-
Educational Studies in Mathematics
46:
13–57, 2001.
©
2002 Kluwer Academic Publishers. Printed in the Netherlands.
14
ANNA SFARD
tribution to both theory and practice of mathematics education. I begin
with taking a close look at two episodes that give rise to a number of
difficult questions. The intricacy of the problems serves as the immediate
motivation for a critical look at traditional cognitive research, based on
the metaphor of learning-as-acquisition, and for the introduction of an
additional conceptual framework, grounded in the metaphor of learning-
as-participation. In the last part of this article, in order to show how the
proposed conceptualization works, I revisit the episodes presented at the
beginning of the paper, reformulate the longstanding questions in the new
language, and re-address the old quandaries with the help of specially

designed analytic tools.
1. QUESTIONS WE HAVE ALWAYS BEEN ASKING ABOUT
MATHEMATICAL THINKING AND ARE STILL WONDERING ABOUT
In spite of its being a relatively young discipline, the study of mathem-
atical thinking has a rich and eventful history. Since its birth in the first
half of the 20
th
century, it has been subject to quite a number of major
shifts (Kilpatrick, 1992; Sfard, 1997). These days it may well be on its
way toward yet another reincarnation. What is it that makes this new field
of research so prone to change? Why is it that mathematics education
researchers never seem truly satisfied with their own past achievements?
There is certainly more than one reason, and I shall deal with some of
them later. For now, let me give a commonsensical answer, likely to be
heard from anybody concerned with mathematics education – teachers,
students, parents, mathematicians, and just ordinary citizens concerned
about the well-being of their children and their society. The immediate
suspect, it seems, is the visible gulf between research and practice, express-
ing itself in the lack of significant, lasting improvement in teaching and
learning that the research is supposed to bring. It seems that there is little
correlation between the intensity of research and research-based develop-
ment in a given country and the average level of performance of mathem-
atics students in this country (see e.g. Macnab, 2000; Schmidt et al., 1999;
Stigler and Hiebert, 1999). This, in turn, means that as researchers we may
have yet a long way to go before our solutions to the most basic prob-
lems asked by frustrated mathematics teachers and by desperate students
become effective in the long run. The issues we are still puzzled about
vary from most general questions regarding our basic assumptions about
mathematical learning, to specific everyday queries occasioned by con-
crete classroom situations. Let me limit myself to just two brief examples

of teachers’ and researchers’ dilemmas.
LOOKING AT THINKING AS COMMUNICATING
15
Example 1: Why do children succeed or fail in mathematical tasks? What
is the nature and the mechanism of the success and of the failure?
Or, better still, why does mathematics seem so very difficult to learn and
why is this learning so prone to failure? This is probably the most obvious
among the frequently asked questions, and it can be formulated at many
different levels. The example that follows provides an opportunity to ob-
serve a ‘failure in the making’ – an unsuccessful attempt at learning that
looks like a rather common everyday occurrence.
Figure 2 shows an excerpt from a conversation between two twelve
year old boys, Ari and Gur, grappling together with one of a long series of
problems supposed to usher them into algebraic thinking and to help them
in learning the notion of function.
1
The boys are dealing with the first
question on the worksheet presented in Figure 1. The question requires
finding the value of the function g(x), represented by a partial table, for the
value of x that does not appear in the table (
g
(6)). Before proceeding, the
reader is advised to take a good look at Ari and Gur’s exchange and try
to answer the most natural questions that come to mind in situation like
this: What can be said about the boys’ understanding from the way they go
about the problem? Does the collaboration contribute in any visible way
to their learning? If either of the students experiences difficulty, what is
the nature of the problem? How could he be helped? What would be an
effective way of overcoming – or preventing altogether – the difficulty he
is facing?

While it is not too hard to answer some of these questions, some oth-
ers seem surprisingly elusive. Indeed, a cursory glance at the transcript
is enough to see that while Ari proceeds smoothly and effectively, Gur
is unable to cope with the task. Moreover, in spite of Ari’s apparently
adequate algebraic skills, the conversation that accompanies the process
of solving does not seem to help Gur. We can conclude by saying that
while Ari’s performance is fully satisfactory, Gur does not ‘pass the test’.
16 ANNA SFARD
LOOKING AT THINKING AS COMMUNICATING
17
So far so good: The basic question about the overall effectiveness of
the students’ problem-solving efforts does not pose any special difficulty.
Our problem begins when we attempt a move beyond this crude evaluation
and venture a quest for a deeper insight into the boys’ thinking. Let us
try, for example, to diagnose the nature of Gur’s difficulty. The first thing
to say would be
“Gur does not understand the concept of function” or,
more precisely, “He does not understand what the formula and the table
are all about, what is their relation, and how they should be used in the
present context”. Although certainly true, this statement has little explan-
atory power. What Tolstoy said about unhappiness seems to be true also
about the lack of understanding: Whoever lacks understanding fails to
understand in his or her own way. We do not know much if we cannot
say anything specific about the unique nature of Gur’s incomprehension.
In tune with a long-standing tradition, many researchers are likely to
approach the problem quite differently. As Davis (
1988
) pointed out, rather
than asking whether a person understands, we should ask how he or she
understands. Indeed, “students usually do deal with meanings”, he says,

except that they often “create their own meanings” (p.

9, emphases in the
original). Thus, we could analyze the event in terms of students’ idiosyn-
cratic conceptual constructions. We could say, for example, that unlike his
partner, Gur has not, as yet, developed an adequate conception of function.
One look at the transcript now, and we identify the familiar nature of the in-
adequacy: The sequence [28]–[34] shows that Gur holds the ill-conceived
idea of linearity, according to which the values of any function should be
proportional to the argument (this belief is a variant of the well known
misconception according to which any function should be linear; see e.g.
Markovitz et al., 1986, Vinner and Dreyfus, 1989).
2
This is important in-
formation, no doubt, but is it enough to satisfy our need for explanation? Is
it enough for us to say we have understood Gur’s thinking? Is it sufficient
to guide us as teachers who wish to help Gur in his learning?
Although endowed with an extensive knowledge of students’ typical
misconceptions, we may still be in the dark about many aspects of this
conversation and, more specifically, about the reasons for Gur’s choices
and responses. Thus, for example, what has been said so far does not give
us a clue about the sources either of Gur’s lasting confusion with the equa-
tion of linear function, or of his inability to follow Ari’s explanations. The
misconception that certainly plays a role in the last part of the exchange
does not account for Gur’s earlier responses to the notion of formula. These
responses seem as unexpected as they are unhelpful. Moreover, although it
is obvious that Gur does struggle for understanding, and although the ideas
he wishes to understand do not appear to be very complex (indeed, what
18
ANNA SFARD

could be more straightforward than the need to substitute a number into the
formula in order to calculate the value of the function for this number?), all
his efforts prove strangely ineffective – they do not seem to take him one
step closer to the understanding of the solution explained time and again
by Ari, It is not easy to decide what kind of action on the part of the ‘more
capable peer’ (Vygotsky, 1978, p.

86) could be of help.
At this point one may claim that the difficulty we are facing as inter-
preters stems mainly from the scarcity of data at hand. The episode we
are looking at does not provide enough information for any decisive state-
ment on Ari’s and Gur’s mathematical thinking, some people are likely
to say. Although certainly true, this claim does not undermine the former
complaint: Although it would certainly be better to have more information,
the episode at hand should also be understood on its own terms. What we
need in order to make sense of the things the two boys are saying in the
given situation are not just additional data, but also, and above all, better
developed ways of looking, organized into more penetrating theories of
mathematical thinking and learning. Before we turn to the story of the cur-
rent quest after such theories, let us look at another case of mathematical
learning.
Example 2: What should count as ‘learning with understanding’?
The notion of understanding, so central to our present deliberations, turns
out to be an inexhaustible source of difficulty for both theorists and practi-
tioners. I will now illustrate this difficulty with yet another example related,
this time, to the famous call for meaningful learning or learning-with-
understanding that has been guiding our instructional policies for many
years. This call was a landmark in the history of educational research in
that it signaled the end of the behaviorist era and the beginning of the new
direction in the study of human cognition. When more than six decades

ago Brownell (1935) issued the exhortation for “full recognition of the
value of children’s experiences” and for making “arithmetic less a chal-
lenge to pupil’s memory and more a challenge to his intelligence” (p. 31),
his words sounded innovative, and even defiant. Eventually, these words
helped to lift the behaviorist ban on the inquiry into the ‘black box’ of
mind. Once the permission to look ‘inside human head’ was given, the
issue of understanding turned into one of the central topics of research.
In spite of the impressive advances of this research, most educators
agree today that finding ways to make the principle of learning-with-under-
standing operative is an extremely difficult task. Methods of ‘meaning-
ful’ teaching “are still not well known, and most mathematics teachers
probably must rely on a set of intuitions about quantitative thinking that
LOOKING AT THINKING AS COMMUNICATING
19
involves both the importance of meaning – however defined – and com-
putation,” complains Mayer (1983, p. 77). Hiebert and Carpenter echo
this concern when saying that promoting learning with understanding “has
been like searching for the Holy Grail.” “There is a persistent belief in
the merits of the goal, but designing school learning environments that
successfully promote learning with understanding has been difficult,” they
add (Hiebert and Carpenter, 1992, p. 65). The conversation between pre-
service teacher Rada and the 7 year old girl Noa about the concept of ‘the
biggest number’ (see Figure 3) highlights a certain aspect of the difficulty.
Clearly, for Noa, this very brief conversation becomes an opportun-
ity for learning. The girl begins the dialogue convinced that there is a
number that can be called ‘the biggest’ and she ends emphatically stating
the opposite: “There is no such number!”. The question is whether this
learning may be regarded as learning-with-understanding, and whether it
is therefore the desirable kind of learning.
To answer this question, one has to look at the way in which the learn-

ing occurs. The seemingly most natural thing to say if one approaches
the task from the traditional perspective, already mentioned in the former
example, is that the teacher leads the girl to realize the contradiction in
her conception of number: Noa views the number set as finite, but she also
seems aware of the fact that adding one to any number leads to an even
bigger number. These two facts, put together, lead to what is called in the
20
ANNA SFARD
literature ‘a cognitive conflict’ (see e.g. Tall and Schwartzenberger, 1978),
and thus call for revision and modification of her number schema. This
is what the girl eventually does. On the face of it, the change occurs as
a result of rational considerations, and may thus count as an instance of
learning with understanding.
And yet, something seems to be missing in this explanation. Why is it
that Noa stays quite unimpressed by the contradiction the first time she is
asked about the number obtained by adding one? Why doesn’t she modify
her answer when exposed to it for the second time? Why is it that when
she eventually puts together the two contradicting claims – the claim that
adding one leads to a bigger number and the claim that there is such thing
as the biggest number – her conclusion ends with a question mark rather
than with a firm assertion (see [22])? Isn’t the girl aware of the logical
necessity of this conclusion?
Another possibility, one I will discuss in detail later in this paper, is
that Noa’s change of mind has less to do with her understanding of the
concepts than with her spontaneous use of mostly involuntary cues about
the appropriateness of her answers found in the teacher’s reactions. In this
case, the decision to say, in the end, that “there is no biggest number” can-
not be regarded as an evidence of ‘learning-with-understanding’, at least
not according to how the term ‘understanding’ is usually interpreted in this
context. If so, the adherents of meaningful learning are likely to criticize

the teacher for the instructional strategy she used. And yet, from my nu-
merous encounters with teachers, I do know, that for the great majority of
them, the way Rada proceeded in the present example would be the natural
choice. Teacher’s intuitions are not anything to be easily dismissed by the
researcher. We seem to be facing yet another dilemma likely to challenge
teachers and researchers.
Summary: On the learning-as-acquisition metaphor, its advantages and
its shortcomings
After having had a look at a number of questions spawned by the two
brief episodes, it is time now to say a few words about research in math-
ematics education in general. The ways researchers have been looking at
the studied phenomena may be diverse and many, but all the known ap-
proaches were, until recently, unified by the same basic vision of learning.
Influenced by folk models of learning implicit in our everyday ways of
talking, and further encouraged by numerous scientific theories of mind
that conceptualize learning as storing information in the form of men-
tal representations, the students of mathematical thinking and problem-
solving tacitly adopted the metaphor of learning as the acquisition of